Turning an urn all blue by replacement

An urn starts with two red balls and one blue ball. On each turn a ball is drawn uniformly at random and, whatever its colour, it is put aside and a blue ball is put in its place. What is the expected number of turns until every ball in the urn is blue?

$9/2$. Each red ball has to be drawn once. With two reds present a draw hits red with probability $\tfrac23$ (mean wait $\tfrac32$); with one red left the chance is $\tfrac13$ (mean wait 3). Adding the geometric waits gives $\tfrac32+3=\tfrac92$.